MATH 74 MIDTERM 1
1(a).
False.
(No matter what
*
is, and no matter what
a
∈
S
is, the equation
a
*
a
=
a
*
a
is
always true, for example.)
1(b).
True.
For any integer
k
, the real number sin(
πk
2
) is an integer: it is either 0, 1, or

1
depending on whether
k
is even, one more than a multiple of four, or one less than
a multiple of four. So the formula for
a
*
b
actually does give an integer result for
any integers
a
and
b
.
1(c).
False.
The problem is that any given rational number has
many
different representa
tions as a quotient
a
b
, and the result of the formula depends on which one you pick.
(I did not specify whether
a, b, c, d
are integers, or are themselves rational numbers;
either way, this is a problem.)
For example, suppose I want to compute
1
2
*
1. If write 1 as
1
1
the formula says
1
2
*
1 is
1
2
. If I write 1 as
13
13
the formula says
1
2
*
1 is
1
26
. Because of this ambiguity,
the given formula does
not
define a binary operation on
Q
.
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 Fall '07
 COURTNEY
 Math, Ring, Natural number, Identity element, sequence equality theorem

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